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Consider two CPTP maps $M_{A\rightarrow B}$ and $N_{A\rightarrow B}$. Let $\Phi = M - N$.

To distinguish between the two maps, there are several measures but here I want to compare two of them. The first is the induced trace norm, which is defined as

$$\|\Phi\|_1 = \sup_{X\in\mathcal{L}(A), \|X\|_1 = 1}\|\Phi(X)\|_1$$

We also have the min-fidelity, which is

$$F_{\min }(\Phi):=\min _{X \in \mathcal{L}(A)} F(\Phi(X), X).$$

Here $\mathcal{L}(A)$ is the set of linear operators in $A$ and $F(\rho, \sigma) = (\operatorname{tr} \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}})^2$ is the fidelity between $\rho$ and $\sigma$.

Are there known bounds that relate these two quantities? I have one using the Fuchs van de Graaf inequalities that relate trace distance and fidelity and a triangle inequality on the trace distance but wondering if anything better can be done.

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